First slide

Extreme values of trigonometric functions in trigonometry

Question

If $θ_{i} > 0 for 1 \leq θ \leq n and θ_{1} + θ_{2} + θ_{3} + \dots + θ_{n} = π,$ then the greatest value of sum $\sin θ_{1} + \sin θ_{2} + \sin θ_{3} + \dots + \sin θ_{n}$ is equal to

Moderate

A

n
B

$nsin (\frac{π}{n})$
C

$π$
D

None of these

Solution

We have,
$\begin{aligned} 0 < i < π, i = 1, 2, 3, \dots n \\ ∴ \sin θ_{i} > 0, i = 1, 2, 3, \dots n \end{aligned}$
$Now, (\frac{\sin θ_{1} + \sin θ_{2} + \sin θ_{3} + \dots + \sin θ_{n}}{n}) \leq \sin (\frac{θ_{1} + θ_{2} + θ_{3} + \dots + θ_{n}}{n}) = \sin (\frac{π}{n})$
$∴ \sin θ_{1} + \sin θ_{2} + \sin θ_{3} + \dots + \sin θ_{n} \leq nsin (\frac{π}{n})$

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